/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files


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MAYBE
proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, INF).

(0) CpxIntTrs
(1) Loat Proof [FINISHED, 3889 ms]
(2) BOUNDS(1, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f0(A, B, C, D, E, F, G) -> Com_1(f10(1, 1, H, 0, 2, 1, G)) :|: H >= 0
f10(A, B, C, D, E, F, G) -> Com_1(f21(A, B - 1, C, D, E, F, 0)) :|: F >= 1 && E >= F && B >= 1 && 0 >= C
f10(A, B, C, D, E, F, G) -> Com_1(f21(A + 1, A + 1, H, D, E, F, I)) :|: I >= 0 && 1 >= I && H >= 0 && F >= 1 && E >= F && 0 >= B && 0 >= C
f10(A, B, C, D, E, F, G) -> Com_1(f21(A, B, C - 1, D, E, F, H)) :|: H >= 0 && 1 >= H && F >= 1 && C >= 1 && E >= F
f21(A, B, C, D, E, F, G) -> Com_1(f10(A, B, C, D, E, F - 1, G)) :|: E + 1 >= A && 0 >= G
f21(A, B, C, D, E, F, G) -> Com_1(f10(A, B, C, D, E, F + 1, G)) :|: E + 1 >= A && G >= 1
f10(A, B, C, D, E, F, G) -> Com_1(f32(A, B, C, D, E, F, G)) :|: 0 >= F && E >= F
f10(A, B, C, D, E, F, G) -> Com_1(f32(A, B, C, D, E, F, G)) :|: F >= 1 + E

The start-symbols are:[f0_7]


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(1) Loat Proof (FINISHED)


### Pre-processing the ITS problem ###



Initial linear ITS problem

   Start location: f0

      0: f0 -> f10 : A'=1, B'=1, C'=free, D'=0, E'=2, F'=1, [ free>=0 ], cost: 1

      1: f10 -> f21 : B'=-1+B, G'=0, [ F>=1 && E>=F && B>=1 && 0>=C ], cost: 1

      2: f10 -> f21 : A'=1+A, B'=1+A, C'=free_2, G'=free_1, [ free_1>=0 && 1>=free_1 && free_2>=0 && F>=1 && E>=F && 0>=B && 0>=C ], cost: 1

      3: f10 -> f21 : C'=-1+C, G'=free_3, [ free_3>=0 && 1>=free_3 && F>=1 && C>=1 && E>=F ], cost: 1

      6: f10 -> f32 : [ 0>=F && E>=F ], cost: 1

      7: f10 -> f32 : [ F>=1+E ], cost: 1

      4: f21 -> f10 : F'=-1+F, [ 1+E>=A && 0>=G ], cost: 1

      5: f21 -> f10 : F'=1+F, [ 1+E>=A && G>=1 ], cost: 1



Removed unreachable and leaf rules:

   Start location: f0

      0: f0 -> f10 : A'=1, B'=1, C'=free, D'=0, E'=2, F'=1, [ free>=0 ], cost: 1

      1: f10 -> f21 : B'=-1+B, G'=0, [ F>=1 && E>=F && B>=1 && 0>=C ], cost: 1

      2: f10 -> f21 : A'=1+A, B'=1+A, C'=free_2, G'=free_1, [ free_1>=0 && 1>=free_1 && free_2>=0 && F>=1 && E>=F && 0>=B && 0>=C ], cost: 1

      3: f10 -> f21 : C'=-1+C, G'=free_3, [ free_3>=0 && 1>=free_3 && F>=1 && C>=1 && E>=F ], cost: 1

      4: f21 -> f10 : F'=-1+F, [ 1+E>=A && 0>=G ], cost: 1

      5: f21 -> f10 : F'=1+F, [ 1+E>=A && G>=1 ], cost: 1



### Simplification by acceleration and chaining ###



Eliminated locations (on tree-shaped paths):

   Start location: f0

      0: f0 -> f10 : A'=1, B'=1, C'=free, D'=0, E'=2, F'=1, [ free>=0 ], cost: 1

      8: f10 -> f10 : B'=-1+B, F'=-1+F, G'=0, [ F>=1 && E>=F && B>=1 && 0>=C && 1+E>=A ], cost: 2

      9: f10 -> f10 : A'=1+A, B'=1+A, C'=free_2, F'=-1+F, G'=free_1, [ free_1>=0 && free_2>=0 && F>=1 && E>=F && 0>=B && 0>=C && 1+E>=1+A && 0>=free_1 ], cost: 2

     10: f10 -> f10 : A'=1+A, B'=1+A, C'=free_2, F'=1+F, G'=free_1, [ 1>=free_1 && free_2>=0 && F>=1 && E>=F && 0>=B && 0>=C && 1+E>=1+A && free_1>=1 ], cost: 2

     11: f10 -> f10 : C'=-1+C, F'=-1+F, G'=free_3, [ free_3>=0 && F>=1 && C>=1 && E>=F && 1+E>=A && 0>=free_3 ], cost: 2

     12: f10 -> f10 : C'=-1+C, F'=1+F, G'=free_3, [ 1>=free_3 && F>=1 && C>=1 && E>=F && 1+E>=A && free_3>=1 ], cost: 2



Accelerating simple loops of location 1.

   Simplified some of the simple loops (and removed duplicate rules).

   Accelerating the following rules:

      8: f10 -> f10 : B'=-1+B, F'=-1+F, G'=0, [ F>=1 && E>=F && B>=1 && 0>=C && 1+E>=A ], cost: 2

      9: f10 -> f10 : A'=1+A, B'=1+A, C'=free_2, F'=-1+F, G'=0, [ free_2>=0 && F>=1 && E>=F && 0>=B && 0>=C && 1+E>=1+A ], cost: 2

     10: f10 -> f10 : A'=1+A, B'=1+A, C'=free_2, F'=1+F, G'=1, [ free_2>=0 && F>=1 && E>=F && 0>=B && 0>=C && 1+E>=1+A ], cost: 2

     11: f10 -> f10 : C'=-1+C, F'=-1+F, G'=0, [ F>=1 && C>=1 && E>=F && 1+E>=A ], cost: 2

     12: f10 -> f10 : C'=-1+C, F'=1+F, G'=1, [ F>=1 && C>=1 && E>=F && 1+E>=A ], cost: 2



   Accelerated rule 8 with metering function F (after adding B>=F), yielding the new rule 13.

   Accelerated rule 8 with metering function B (after adding B<=F), yielding the new rule 14.

   Found no metering function for rule 9.

   Found no metering function for rule 10.

   Accelerated rule 11 with metering function F (after adding C>=F), yielding the new rule 15.

   Accelerated rule 11 with metering function C (after adding C<=F), yielding the new rule 16.

   Accelerated rule 12 with backward acceleration, yielding the new rule 17.

   Accelerated rule 12 with backward acceleration, yielding the new rule 18.

      During metering: Instantiating temporary variables by {free_2==1}

      During metering: Instantiating temporary variables by {free_2==1}

   Removing the simple loops: 8 11 12.



Accelerated all simple loops using metering functions (where possible):

   Start location: f0

      0: f0 -> f10 : A'=1, B'=1, C'=free, D'=0, E'=2, F'=1, [ free>=0 ], cost: 1

      9: f10 -> f10 : A'=1+A, B'=1+A, C'=free_2, F'=-1+F, G'=0, [ free_2>=0 && F>=1 && E>=F && 0>=B && 0>=C && 1+E>=1+A ], cost: 2

     10: f10 -> f10 : A'=1+A, B'=1+A, C'=free_2, F'=1+F, G'=1, [ free_2>=0 && F>=1 && E>=F && 0>=B && 0>=C && 1+E>=1+A ], cost: 2

     13: f10 -> f10 : B'=-F+B, F'=0, G'=0, [ F>=1 && E>=F && B>=1 && 0>=C && 1+E>=A && B>=F ], cost: 2*F

     14: f10 -> f10 : B'=0, F'=F-B, G'=0, [ F>=1 && E>=F && B>=1 && 0>=C && 1+E>=A && B<=F ], cost: 2*B

     15: f10 -> f10 : C'=-F+C, F'=0, G'=0, [ F>=1 && C>=1 && E>=F && 1+E>=A && C>=F ], cost: 2*F

     16: f10 -> f10 : C'=0, F'=F-C, G'=0, [ F>=1 && C>=1 && E>=F && 1+E>=A && C<=F ], cost: 2*C

     17: f10 -> f10 : C'=0, F'=F+C, G'=1, [ F>=1 && C>=1 && E>=F && 1+E>=A && -1+F+C>=1 && E>=-1+F+C ], cost: 2*C

     18: f10 -> f10 : C'=-1+F+C-E, F'=1+E, G'=1, [ F>=1 && C>=1 && E>=F && 1+E>=A && E>=1 && F+C-E>=1 ], cost: 2-2*F+2*E



Chained accelerated rules (with incoming rules):

   Start location: f0

      0: f0 -> f10 : A'=1, B'=1, C'=free, D'=0, E'=2, F'=1, [ free>=0 ], cost: 1

     19: f0 -> f10 : A'=1, B'=0, C'=0, D'=0, E'=2, F'=0, G'=0, [], cost: 3

     20: f0 -> f10 : A'=1, B'=0, C'=0, D'=0, E'=2, F'=0, G'=0, [], cost: 3

     21: f0 -> f10 : A'=1, B'=1, C'=-1+free, D'=0, E'=2, F'=0, G'=0, [ free>=1 ], cost: 3

     22: f0 -> f10 : A'=1, B'=1, C'=0, D'=0, E'=2, F'=0, G'=0, [], cost: 3

     23: f0 -> f10 : A'=1, B'=1, C'=0, D'=0, E'=2, F'=1+free, G'=1, [ free>=1 && 2>=free ], cost: 1+2*free

     24: f0 -> f10 : A'=1, B'=1, C'=-2+free, D'=0, E'=2, F'=3, G'=1, [ -1+free>=1 ], cost: 5



Removed unreachable locations (and leaf rules with constant cost):

   Start location: f0

     23: f0 -> f10 : A'=1, B'=1, C'=0, D'=0, E'=2, F'=1+free, G'=1, [ free>=1 && 2>=free ], cost: 1+2*free



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f0

     23: f0 -> f10 : A'=1, B'=1, C'=0, D'=0, E'=2, F'=1+free, G'=1, [ free>=1 && 2>=free ], cost: 1+2*free



Computing asymptotic complexity for rule 23

   Could not solve the limit problem.

   Resulting cost 0 has complexity: Unknown



Obtained the following overall complexity (w.r.t. the length of the input n):

   Complexity:  Unknown

   Cpx degree:  ?

   Solved cost: 0

   Rule cost:   0

   Rule guard:  []



WORST_CASE(Omega(0),?)


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(2)
BOUNDS(1, INF)